Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

and consider functions . Then, both integrals

       exists, whenever one of the integrals exists, in which case, we have

      1.3.4 The Fundamental Theorem of Calculus

      The first result we present in this section is the Fundamental Theorem of Calculus for the variational Henstock integral. The proof follows standard steps (see [172], p. 43, for instance) adapted to Banach space-valued functions.

      Theorem 1.73 (Fundamental Theorem of Calculus): Suppose is a function such that there exists the derivative , for every . Then, and

Next, we give an example, borrowed from [73], of a Banach space-valued function which is Riemann–McShane integrable (see the appendix to this chapter for this definition). However, is not variationally Henstock integrable, nor it is integrable in the sense of Bochner–Lebesgue.

      Example 1.74: Let and consider the function given by , where denotes the characteristic function of a measurable set .

      Since (see Definition 1.24) and the function , , is an element of , the abstract Riemann–Stieltjes integral, , exists (see [127, Theorem 4.6], p. 24). Moreover, the Riemann–Stieltjes integral, , also exists and the integration by parts formula

      holds (see Theorem 1.53). Hence, .

      Consider the indefinite integral , , of . Then,

      and, hence,

      Thus, is absolutely continuous.

      On the other hand, is nowhere differentiable (see [73], Example 3.1). Then, the Lebesgue theorem implies , where by , we denote the space of functions from